They want you to minimize

    f(x,y) = (x-6)*(y-4)  under the condition  x*y = 96.  (1)

where x and y are the dimensions of the poster.


In other words, from (1), you need to minimize

    f(x,y) = xy - 4x - 6y + 24 = 96 - 4x - 6y + 24 = -4x - 6y + 120.


From (1), you have  y = , so the function f(x,y) takes the form

    g(x) = -4x -  + 120 = -4x -  + 120.    (2)


So, you differentiate (2), and you get

    g'(x) = -4 + .


Equate it to zero

    g'(x) = 0 = -4 + .


So, to find x, you have this equation

     = 576,

     =  = 144

     x =  = 12 centimeters.


Then y =  = 8 cm.


Thus the dimension of the poster are  12 inches width and 8 inches height.